Friedrich Ludwig Gottlob Frege (Template:IPA-de; 8 November 1848 – 26 July 1925) was a Germanmathematician, logician and philosopher. He is considered to be one of the founders of modern logic and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on the philosophy of language and mathematics. While he was mainly ignored by the intellectual world when he published his writings, Giuseppe Peano (1858–1932) and Bertrand Russell (1872–1970) introduced his work to later generations of logicians and philosophers.

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Frege was born in 1848 in Wismar, in the state of Mecklenburg-Schwerin (the modern German federal state Mecklenburg-Vorpommern). His father Carl (Karl) Alexander Frege (3 August 1809 – 30 November 1866) was the co-founder and headmaster of a girls' high school until his death. After Carl's death, the school was led by Frege's mother Auguste Wilhelmine Sophie Frege (née Bialloblotzky of Polish descent, 12 January 1815 – 14 October 1898).

In childhood, Frege encountered philosophies that would guide his future scientific career. For example, his father wrote a textbook on the German language for children aged 9–13, entitled Hülfsbuch zum Unterrichte in der deutschen Sprache für Kinder von 9 bis 13 Jahren (2nd ed., Wismar, 1850; 3rd ed., Wismar and Ludwigslust: Hinstorff, 1862), the first section of which dealt with the structure and logic of language.

Frege studied at a gymnasium in Wismar and graduated in 1869. His teacher Gustav Adolf Leo Sachse (5 November 1843 – 1 September 1909), who was also a poet, played the most important role in determining Frege's future scientific career, encouraging him to continue his studies at the University of Jena.

Frege matriculated at the University of Jena in the spring of 1869 as a citizen of the North German Federation. In the four semesters of his studies he attended approximately twenty courses of lectures, most of them on mathematics and physics. His most important teacher was Ernst Karl Abbe (1840–1905) (physicist, mathematician, and inventor). Abbe gave lectures on theory of gravity galvanism and electrodynamics, complex analysis theory of functions of a complex variable, applications of physics, selected divisions of mechanics, and mechanics of solids. Abbe was more than a teacher to Frege: he was a trusted friend, and, as director of the optical manufacturer Carl Zeiss AG, he was in a position to advance Frege's career. After Frege's graduation, they came into closer correspondence.

In 1873, Frege attained his doctorate under Ernst Schering, with a dissertation under the title of "Über eine geometrische Darstellung der imaginären Gebilde in der Ebene" ("On a Geometrical Representation of Imaginary Forms in a Plane"), in which he aimed to solve such fundamental problems in geometry as the mathematical interpretation of projective geometry's infinitely distant (imaginary) points.

Though his education and early work were mathematical, especially geometrical, Frege's thought soon turned to logic. His Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (Halle a/S: Verlag von Louis Nebert, 1879) (Concept-Script: A Formal Language for Pure Thought Modeled on that of Arithmetic) marked a turning point in the history of logic. The Begriffsschrift broke new ground, including a rigorous treatment of the ideas of functions and variables. Frege wanted to show that mathematics grows out of logic, but in so doing, he devised techniques that took him far beyond the Aristotelian syllogistic and Stoic propositional logic that had come down to him in the logical tradition.File:Begriffsschrift Titel.png In effect, Frege invented axiomaticpredicate logic, in large part thanks to his invention of quantified variables, which eventually became ubiquitous in mathematics and logic, and which solved the problem of multiple generality. Previous logic had dealt with the logical constantsand, or, if ... then ..., not, and some and all, but iterations of these operations, especially "some" and "all", were little understood: even the distinction between a pair of sentences like "every boy loves some girl" and "some girl is loved by every boy" was able to be represented only very artificially, whereas Frege's formalism had no difficulty expressing the different readings of "every boy loves some girl who loves some boy who loves some girl" and similar sentences, in complete parallel with his treatment of, say, "every boy is foolish".

One of Frege's stated purposes was to isolate genuinely logical principles of inference, so that in the proper representation of mathematical proof, one would at no point appeal to "intuition". If there was an intuitive element, it was to be isolated and represented separately as an axiom: from there on, the proof was to be purely logical and without gaps. Having exhibited this possibility, Frege's larger purpose was to defend the view that arithmetic is a branch of logic, a view known as logicism: unlike geometry, arithmetic was to be shown to have no basis in "intuition", and no need for non-logical axioms. Already in the 1879 Begriffsschrift important preliminary theorems, for example a generalized form of law of trichotomy, were derived within what Frege understood to be pure logic.

This idea was formulated in non-symbolic terms in his Die Grundlagen der Arithmetik (1884) (The Foundations of Arithmetic). Later, in his Grundgesetze der Arithmetik (Basic Laws of Arithmetic) (vol. 1, 1893; vol. 2, 1903) (vol. 2 of which was published at his own expense), Frege attempted to derive, by use of his symbolism, all of the laws of arithmetic from axioms he asserted as logical. Most of these axioms were carried over from his Begriffsschrift, though not without some significant changes. The one truly new principle was one he called the Basic Law V: the "value-range" of the function f(x) is the same as the "value-range" of the function g(x) if and only if ∀x[f(x) = g(x)].

The crucial case of the law may be formulated in modern notation as follows. Let {x|Fx} denote the extension of the predicateFx, i.e., the set of all Fs, and similarly for Gx. Then Basic Law V says that the predicates Fx and Gx have the same extension iff ∀x[Fx ↔ Gx]. The set of Fs is the same as the set of Gs just in case every F is a G and every G is an F. (The case is special because what is here being called the extension of a predicate, or a set, is only one type of "value-range" of a function.)

In a famous episode, Bertrand Russell wrote to Frege, just as Vol. 2 of the Grundgesetze was about to go to press in 1903, showing that Russell's paradox could be derived from Frege's Basic Law V. It is easy to define the relation of membership of a set or extension in Frege's system; Russell then drew attention to "the set of things x that are such that x is not a member of x". The system of the Grundgesetze entails that the set thus characterised both is and is not a member of itself, and is thus inconsistent. Frege wrote a hasty, last-minute Appendix to Vol. 2, deriving the contradiction and proposing to eliminate it by modifying Basic Law V. Frege opened the Appendix with the exceptionally honest comment: "Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion." (This letter and Frege's reply are translated in Jean van Heijenoort 1967.)

Frege's proposed remedy was subsequently shown to imply that there is but one object in the universe of discourse, and hence is worthless (indeed, this would make for a contradiction in Frege's system if he had axiomatized the idea, fundamental to his discussion, that the True and the False are distinct objects; see, for example, Dummett 1973), but recent work has shown that much of the program of the Grundgesetze might be salvaged in other ways:

Basic Law V can be weakened in other ways. The best-known way is due to philosopher and mathematical logician George Boolos (1940–1996), who was an expert on the work of Frege. A "concept" F is "small" if the objects falling under F cannot be put into one-to-one correspondence with the universe of discourse, that is, if: ∃R[R is 1-to-1 & ∀x∃y(xRy & Fy)]. Now weaken V to V*: a "concept" F and a "concept" G have the same "extension" if and only if neither F nor G is small or ∀x(Fx ↔ Gx). V* is consistent if second-order arithmetic is, and suffices to prove the axioms of second-order arithmetic.

Basic Law V can simply be replaced with Hume's Principle, which says that the number of Fs is the same as the number of Gs if and only if the Fs can be put into a one-to-one correspondence with the Gs. This principle, too, is consistent if second-order arithmetic is, and suffices to prove the axioms of second-order arithmetic. This result is termed Frege's Theorem because it was noticed that in developing arithmetic, Frege's use of Basic Law V is restricted to a proof of Hume's Principle; it is from this, in turn, that arithmetical principles are derived. On Hume's Principle and Frege's Theorem, see "Frege's Logic, Theorem, and Foundations for Arithmetic".[2]

Frege's logic, now known as second-order logic, can be weakened to so-called predicative second-order logic. Predicative second-order logic plus Basic Law V is provably consistent by finitistic or constructive methods, but it can interpret only very weak fragments of arithmetic.[3]

Frege's work in logic had little international attention until 1903 when Russell wrote an appendix to The Principles of Mathematics stating his differences with Frege. The diagrammatic notation
that Frege used had no antecedents (and has had no imitators since). Moreover, until Russell and Whitehead's Principia Mathematica (3 vols.) appeared in 1910–13, the dominant approach to mathematical logic was still that of George Boole (1815–1864) and his intellectual descendants, especially Ernst Schröder (1841–1902). Frege's logical ideas nevertheless spread through the writings of his student Rudolf Carnap (1891–1970) and other admirers, particularly Bertrand Russell and Ludwig Wittgenstein (1889–1951).

As a philosopher of mathematics, Frege attacked the psychologistic appeal to mental explanations of the content of judgment of the meaning of sentences. His original purpose was very far from answering general questions about meaning; instead, he devised his logic to explore the foundations of arithmetic, undertaking to answer questions such as "What is a number?" or "What objects do number-words ("one", "two", etc.) refer to?" But in pursuing these matters, he eventually found himself analysing and explaining what meaning is, and thus came to several conclusions that proved highly consequential for the subsequent course of analytic philosophy and the philosophy of language.

It should be kept in mind that Frege was employed as a mathematician, not a philosopher, and he published his philosophical papers in scholarly journals that often were hard to access outside of the German-speaking world. He never published a philosophical monograph other than The Foundations of Arithmetic, much of which was mathematical in content, and the first collections of his writings appeared only after World War II. A volume of English translations of Frege's philosophical essays first appeared in 1952, edited by students of Wittgenstein, Peter Geach (born 1916) and Max Black (1909–1988), with the bibliographic assistance of Wittgenstein (see Geach, ed. 1975, Introduction). Despite the generous praise of Russell and Wittgenstein, Frege was little known as a philosopher during his lifetime. His ideas spread chiefly through those he influenced, such as Russell, Wittgenstein, and Carnap, and through work on logic and semantics by Polish logicians.

The distinction between Sinn ("sense") and Bedeutung (usually translated "reference", but also as "meaning" or "denotation") was an innovation of Frege in his 1892 paper "Über Sinn und Bedeutung" ("On Sense and Reference"). According to Frege, sense and reference are two different aspects of the significance of an expression. Frege applied Bedeutung in the first instance to proper names, where it means the bearer of the name, the object in question, but then also to other expressions, including complete sentences, which bedeuten the two "truth values", the true and the false; by contrast, the sense or Sinn associated with a complete sentence is the thought it expresses. The sense of an expression is said to be the "mode of presentation" of the item referred to.

The distinction can be illustrated thus: In their ordinary uses, the name "Charles Philip Arthur George Mountbatten-Windsor", which for logical purposes is an unanalyzable whole, and the functional expression "the Prince of Wales", which contains the significant parts "the prince of ξ" and "Wales", have the same reference, namely, the person best known as Prince Charles. But the sense of the word "Wales" is a part of the sense of the latter expression, but no part of the sense of the "full name" of Prince Charles.

Imagine the road signs outside a city. They all point to (bedeuten) the same object (the city), although the "mode of presentation" or sense (Sinn) of each sign (its direction or distance) is different. Similarly "the Prince of Wales" and "Charles Philip Arthur George Mountbatten-Windsor" both denote (bedeuten) the same object, though each uses a different "mode of presentation" (sense or Sinn).

Frege's published philosophical writings were of a very technical nature and divorced from practical issues, so much so that Frege scholar Dummett expresses his "shock to discover, while reading Frege's diary, that his hero was an outspoken anti-Semite (1973)."[4] He was always a conservative, but after World War I he became more of a radical. His late political "diary shows Frege to have been a man of extreme right-wing political opinions, bitterly opposed to the parliamentary system, democrats, liberals, Catholics, the French and, above all, Jews, who he thought ought to be deprived of political rights and, preferably, expelled from Germany".[5] Frege confided "that he had once thought of himself as a liberal and was an admirer of Bismarck, but his heroes now were General Ludendorff and Adolf Hitler. This was after the two had tried to topple the elected democratic government in a coup in November 1923. In his diary Frege also used all his analytic skills to devise plans for expelling the Jews from Germany and for suppressing the Social Democrats."[6] Frege disliked universal suffrage and was against any form of socialism, which he simply called Marxism. His antisemitism still allowed for exceptions, and he had friendly relations with Jews in real life: among his students was Gershom Scholem who much valued his teacher;[7][8] and he encouraged Ludwig Wittgenstein to leave for England. The 1924 diary has been published.[9]

Logical Investigations (1918–1923).
Frege intended that the following three papers be published together in a book titled Logische Untersuchungen (Logical Investigations). Though the German book never appeared, English translations did appear together in Logical Investigations, ed. Peter Geach, Blackwell's, 1975.